Laws of large numbers for ratios of uniform random variables

Abstract

Let fX n ; n 1g and fY n ; n 1g be two sequences of uniform random variables. We obtain various strong and weak laws of large numbers for the ratio of these two sequences. Even though these are uniform and naturally bounded random variables the ratios are not bounded and have an unusual behaviour creating Exact Strong Laws.

On Exact Laws of Large Numbers for Oppenheim Expansions with Infinite Mean

“…Exact weak laws for i.i.d. random variables can be found in [3], [4] and [19], while the assumption of identically distributed random variables is dropped in [5]. Exact weak laws of large numbers can also be found in the literature for dependent random variables (see for example [16] and [22]).…”

Limit theorems for ratios of order statistics from uniform distributions

“…In this case, the result is independent of the sample size m n because the density of R n12 is independent of m n . Thus, there is no difference between the assumption m n → ∞ and for fixed m n = m. Furthermore, if we take b n = (log n) α+2 and a n = (log n) α /n, then for α > -2, the conditions of Theorem 3.1 hold and λ = 1/(α + 2), demonstrating that Theorem 3.1 in Adler [1] and Theorem 2.3 in Miao et al [7] are special cases of Theorem 3.1.…”

“…In this paper, some generalized results of {R nij , n ≥ 1} for the fixed sample size m n = m are established based on the works by Adler [1], Miao et al [7], and Xu and Miao [11]. In Sect.…”

Strong laws for the largest ratio of adjacent order statistics

“…[December of these order statistics from the uniform distribution was explored in [3], then later in [7]. The exponential distribution was first covered in [4], then in [6] and also in [8].…”